Golden Ratio Proximal Gradient ADMM for Distributed Composite Convex Optimization

This paper introduces a golden ratio proximal gradient alternating direction method of multipliers (GRPG-ADMM) for distributed composite convex optimization. When applied to the consensus optimization problem, the GRPG-ADMM provides a superior safety protection approach to computing an optimal decision for a network of agents connected by edges in an undirected graph. To ensure convergence, we expand the parameter range used in the convex combination step. The algorithm has been implemented to solve a decentralized composite convex consensus optimization problem, resulting in a single-loop decentralized golden ratio proximal gradient algorithm. Notably, the agents do not need to communicate with their neighbors before launching the algorithm. Our algorithm guarantees convergence for both primal and dual iterates, with an ergodic convergence rate of $$\mathcal O(1/k)$$ O ( 1 / k ) measured by function value residual and consensus violation. To demonstrate the efficiency of the algorithm, we compare its performance with two state-of-the-art algorithms and present numerical results on the decentralized sparse group LASSO problem and the decentralized compressed sensing problem.